Derivatives of meromorphic functions and sine function

Abstract

In the paper, we take up a new method to prove the following result. Let $f$ be a meromorphic function in the complex plane, all of whose zeros have multiplicity at least $k+1$ ($k\geq 2$) and all of whose poles are multiple. If $T(r,\sin z)=o\{T(r,f(z))\}$ as $n\rightarrow\infty$, then $f^{(k)}(z)-\sin z$ has infinitely many zeros.